LMFDB - Embedded newform 2160.3.c.o.1889.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 76x^{10} + 1798x^{8} + 15824x^{6} + 40465x^{4} + 25444x^{2} + 196 $ x^12 + 76x^10 + 1798x^8 + 15824x^6 + 40465x^4 + 25444*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.9
Root			$-3.85006i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.o.1889.10

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.03477 - 2.95308i) q^{5} +7.07219i q^{7} +9.43298i q^{11} -6.86548i q^{13} -6.05971 q^{17} +14.3359 q^{19} +11.4980 q^{23} +(7.55866 - 23.8300i) q^{25} -21.2098i q^{29} +8.47239 q^{31} +(20.8847 + 28.5346i) q^{35} +22.0334i q^{37} +58.8487i q^{41} -49.4038i q^{43} +49.9243 q^{47} -1.01582 q^{49} -69.7104 q^{53} +(27.8563 + 38.0599i) q^{55} +50.0747i q^{59} +92.9186 q^{61} +(-20.2743 - 27.7006i) q^{65} +42.1261i q^{67} +84.2630i q^{71} +108.713i q^{73} -66.7118 q^{77} -23.2680 q^{79} +55.7104 q^{83} +(-24.4495 + 17.8948i) q^{85} -73.1674i q^{89} +48.5540 q^{91} +(57.8419 - 42.3349i) q^{95} +69.4827i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{5} + 6 q^{25} - 12 q^{31} + 30 q^{35} - 48 q^{47} - 24 q^{49} + 60 q^{53} - 30 q^{55} - 120 q^{61} + 108 q^{65} - 204 q^{77} - 72 q^{79} + 84 q^{83} - 84 q^{85} - 48 q^{91} + 96 q^{95}+O(q^{100}) $ 12 * q - 6 * q^5 + 6 * q^25 - 12 * q^31 + 30 * q^35 - 48 * q^47 - 24 * q^49 + 60 * q^53 - 30 * q^55 - 120 * q^61 + 108 * q^65 - 204 * q^77 - 72 * q^79 + 84 * q^83 - 84 * q^85 - 48 * q^91 + 96 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.03477	−	2.95308i	0.806953	−	0.590616i
$5$	4.03477	−	2.95308i	0.806953	−	0.590616i
$6$		0			0
$7$			7.07219i			1.01031i	0.863028	+	0.505156i	$0.168565\pi$
$7$			7.07219i			1.01031i	−0.863028	+	0.505156i	$0.831435\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.43298i			0.857543i	0.903413	+	0.428772i	$0.141054\pi$
$11$			9.43298i			0.857543i	−0.903413	+	0.428772i	$0.858946\pi$
$12$		0			0
$13$		−	6.86548i		−	0.528114i	−0.964507	−	0.264057i	$-0.914939\pi$
$13$		−	6.86548i		−	0.528114i	0.964507	−	0.264057i	$-0.0850607\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.05971			−0.356454			−0.178227	−	0.983989i	$-0.557036\pi$
$17$	−6.05971			−0.356454			−0.178227	+	0.983989i	$0.557036\pi$
$18$		0			0
$19$	14.3359			0.754519			0.377260	−	0.926108i	$-0.376866\pi$
$19$	14.3359			0.754519			0.377260	+	0.926108i	$0.376866\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	11.4980			0.499914			0.249957	−	0.968257i	$-0.419583\pi$
$23$	11.4980			0.499914			0.249957	+	0.968257i	$0.419583\pi$
$24$		0			0
$25$	7.55866	−	23.8300i	0.302347	−	0.953198i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	21.2098i		−	0.731373i	−0.930738	−	0.365687i	$-0.880834\pi$
$29$		−	21.2098i		−	0.731373i	0.930738	−	0.365687i	$-0.119166\pi$
$30$		0			0
$31$	8.47239			0.273303			0.136651	−	0.990619i	$-0.456366\pi$
$31$	8.47239			0.273303			0.136651	+	0.990619i	$0.456366\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	20.8847	+	28.5346i	0.596706	+	0.815275i
$36$		0			0
$37$			22.0334i			0.595497i	0.954644	+	0.297748i	$0.0962356\pi$
$37$			22.0334i			0.595497i	−0.954644	+	0.297748i	$0.903764\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			58.8487i			1.43533i	0.696387	+	0.717667i	$0.254790\pi$
$41$			58.8487i			1.43533i	−0.696387	+	0.717667i	$0.745210\pi$
$42$		0			0
$43$		−	49.4038i		−	1.14893i	−0.818530	−	0.574463i	$-0.805211\pi$
$43$		−	49.4038i		−	1.14893i	0.818530	−	0.574463i	$-0.194789\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	49.9243			1.06222			0.531110	−	0.847303i	$-0.321775\pi$
$47$	49.9243			1.06222			0.531110	+	0.847303i	$0.321775\pi$
$48$		0			0
$49$	−1.01582			−0.0207310
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−69.7104			−1.31529			−0.657645	−	0.753328i	$-0.728447\pi$
$53$	−69.7104			−1.31529			−0.657645	+	0.753328i	$0.728447\pi$
$54$		0			0
$55$	27.8563	+	38.0599i	0.506478	+	0.691997i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			50.0747i			0.848723i	0.905493	+	0.424362i	$0.139501\pi$
$59$			50.0747i			0.848723i	−0.905493	+	0.424362i	$0.860499\pi$
$60$		0			0
$61$	92.9186			1.52326			0.761628	−	0.648015i	$-0.224401\pi$
$61$	92.9186			1.52326			0.761628	+	0.648015i	$0.224401\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−20.2743	−	27.7006i	−0.311912	−	0.426163i
$66$		0			0
$67$			42.1261i			0.628747i	0.949299	+	0.314374i	$0.101794\pi$
$67$			42.1261i			0.628747i	−0.949299	+	0.314374i	$0.898206\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			84.2630i			1.18680i	0.804907	+	0.593402i	$0.202215\pi$
$71$			84.2630i			1.18680i	−0.804907	+	0.593402i	$0.797785\pi$
$72$		0			0
$73$			108.713i			1.48922i	0.667499	+	0.744611i	$0.267365\pi$
$73$			108.713i			1.48922i	−0.667499	+	0.744611i	$0.732635\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−66.7118			−0.866387
$78$		0			0
$79$	−23.2680			−0.294532			−0.147266	−	0.989097i	$-0.547047\pi$
$79$	−23.2680			−0.294532			−0.147266	+	0.989097i	$0.547047\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	55.7104			0.671210			0.335605	−	0.942003i	$-0.391059\pi$
$83$	55.7104			0.671210			0.335605	+	0.942003i	$0.391059\pi$
$84$		0			0
$85$	−24.4495	+	17.8948i	−0.287642	+	0.210527i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	73.1674i		−	0.822105i	−0.911612	−	0.411053i	$-0.865161\pi$
$89$		−	73.1674i		−	0.822105i	0.911612	−	0.411053i	$-0.134839\pi$
$90$		0			0
$91$	48.5540			0.533560
$92$		0			0
$93$		0			0
$94$		0			0
$95$	57.8419	−	42.3349i	0.608862	−	0.445631i
$96$		0			0
$97$			69.4827i			0.716317i	0.933661	+	0.358158i	$0.116595\pi$
$97$			69.4827i			0.716317i	−0.933661	+	0.358158i	$0.883405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	128.367i		−	1.27096i	−0.772116	−	0.635482i	$-0.780801\pi$
$101$		−	128.367i		−	1.27096i	0.772116	−	0.635482i	$-0.219199\pi$
$102$		0			0
$103$			52.7014i			0.511664i	0.966721	+	0.255832i	$0.0823494\pi$
$103$			52.7014i			0.511664i	−0.966721	+	0.255832i	$0.917651\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.20204			−0.0579630			−0.0289815	−	0.999580i	$-0.509226\pi$
$107$	−6.20204			−0.0579630			−0.0289815	+	0.999580i	$0.509226\pi$
$108$		0			0
$109$	61.4661			0.563909			0.281955	−	0.959428i	$-0.409017\pi$
$109$	61.4661			0.563909			0.281955	+	0.959428i	$0.409017\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.4323			−0.127719			−0.0638596	−	0.997959i	$-0.520341\pi$
$113$	−14.4323			−0.127719			−0.0638596	+	0.997959i	$0.520341\pi$
$114$		0			0
$115$	46.3918	−	33.9546i	0.403407	−	0.295257i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	42.8554i		−	0.360130i
$120$		0			0
$121$	32.0189			0.264619
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−39.8743	−	118.470i	−0.318994	−	0.947757i
$126$		0			0
$127$			217.236i			1.71052i	0.518197	+	0.855261i	$0.326604\pi$
$127$			217.236i			1.71052i	−0.518197	+	0.855261i	$0.673396\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			116.241i			0.887334i	0.896192	+	0.443667i	$0.146323\pi$
$131$			116.241i			0.887334i	−0.896192	+	0.443667i	$0.853677\pi$
$132$		0			0
$133$			101.386i			0.762300i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−131.556			−0.960264			−0.480132	−	0.877196i	$-0.659411\pi$
$137$	−131.556			−0.960264			−0.480132	+	0.877196i	$0.659411\pi$
$138$		0			0
$139$	246.744			1.77513			0.887567	−	0.460679i	$-0.152394\pi$
$139$	246.744			1.77513			0.887567	+	0.460679i	$0.152394\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	64.7620			0.452881
$144$		0			0
$145$	−62.6343	−	85.5767i	−0.431960	−	0.590184i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	177.624i		−	1.19211i	−0.802945	−	0.596054i	$-0.796735\pi$
$149$		−	177.624i		−	1.19211i	0.802945	−	0.596054i	$-0.203265\pi$
$150$		0			0
$151$	−170.603			−1.12982			−0.564912	−	0.825151i	$-0.691090\pi$
$151$	−170.603			−1.12982			−0.564912	+	0.825151i	$0.691090\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	34.1841	−	25.0196i	0.220543	−	0.161417i
$156$		0			0
$157$			19.7206i			0.125609i	0.998026	+	0.0628046i	$0.0200045\pi$
$157$			19.7206i			0.125609i	−0.998026	+	0.0628046i	$0.979996\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			81.3162i			0.505069i
$162$		0			0
$163$			205.296i			1.25949i	0.776804	+	0.629743i	$0.216840\pi$
$163$			205.296i			1.25949i	−0.776804	+	0.629743i	$0.783160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	77.6781			0.465138			0.232569	−	0.972580i	$-0.425287\pi$
$167$	77.6781			0.465138			0.232569	+	0.972580i	$0.425287\pi$
$168$		0			0
$169$	121.865			0.721095
$170$		0			0
$171$		0			0
$172$		0			0
$173$	35.6959			0.206334			0.103167	−	0.994664i	$-0.467102\pi$
$173$	35.6959			0.206334			0.103167	+	0.994664i	$0.467102\pi$
$174$		0			0
$175$	168.530	+	53.4563i	0.963028	+	0.305464i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	276.264i		−	1.54338i	−0.636001	−	0.771688i	$-0.719412\pi$
$179$		−	276.264i		−	1.54338i	0.636001	−	0.771688i	$-0.280588\pi$
$180$		0			0
$181$	287.557			1.58871			0.794357	−	0.607451i	$-0.207808\pi$
$181$	287.557			1.58871			0.794357	+	0.607451i	$0.207808\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	65.0663	+	88.8995i	0.351710	+	0.480538i
$186$		0			0
$187$		−	57.1612i		−	0.305675i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			63.4669i			0.332288i	0.986102	+	0.166144i	$0.0531316\pi$
$191$			63.4669i			0.332288i	−0.986102	+	0.166144i	$0.946868\pi$
$192$		0			0
$193$		−	238.164i		−	1.23401i	−0.786960	−	0.617004i	$-0.788346\pi$
$193$		−	238.164i		−	1.23401i	0.786960	−	0.617004i	$-0.211654\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−131.777			−0.668918			−0.334459	−	0.942410i	$-0.608554\pi$
$197$	−131.777			−0.668918			−0.334459	+	0.942410i	$0.608554\pi$
$198$		0			0
$199$	310.456			1.56008			0.780041	−	0.625729i	$-0.215198\pi$
$199$	310.456			1.56008			0.780041	+	0.625729i	$0.215198\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	150.000			0.738915
$204$		0			0
$205$	173.785	+	237.441i	0.847730	+	1.15825i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			135.230i			0.647033i
$210$		0			0
$211$	71.4570			0.338659			0.169329	−	0.985560i	$-0.445840\pi$
$211$	71.4570			0.338659			0.169329	+	0.985560i	$0.445840\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−145.893	−	199.333i	−0.678574	−	0.927130i
$216$		0			0
$217$			59.9183i			0.276121i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			41.6029i			0.188248i
$222$		0			0
$223$		−	199.600i		−	0.895066i	−0.894267	−	0.447533i	$-0.852303\pi$
$223$		−	199.600i		−	0.895066i	0.894267	−	0.447533i	$-0.147697\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−316.068			−1.39237			−0.696185	−	0.717863i	$-0.745121\pi$
$227$	−316.068			−1.39237			−0.696185	+	0.717863i	$0.745121\pi$
$228$		0			0
$229$	−27.4174			−0.119727			−0.0598634	−	0.998207i	$-0.519066\pi$
$229$	−27.4174			−0.119727			−0.0598634	+	0.998207i	$0.519066\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−27.5604			−0.118285			−0.0591425	−	0.998250i	$-0.518837\pi$
$233$	−27.5604			−0.118285			−0.0591425	+	0.998250i	$0.518837\pi$
$234$		0			0
$235$	201.433	−	147.430i	0.857161	−	0.627363i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			462.914i			1.93688i	0.249247	+	0.968440i	$0.419817\pi$
$239$			462.914i			1.93688i	−0.249247	+	0.968440i	$0.580183\pi$
$240$		0			0
$241$	438.560			1.81975			0.909876	−	0.414880i	$-0.136176\pi$
$241$	438.560			1.81975			0.909876	+	0.414880i	$0.136176\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.09859	+	2.99979i	−0.0167289	+	0.0122441i
$246$		0			0
$247$		−	98.4227i		−	0.398472i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			102.340i			0.407728i	0.978999	+	0.203864i	$0.0653500\pi$
$251$			102.340i			0.407728i	−0.978999	+	0.203864i	$0.934650\pi$
$252$		0			0
$253$			108.461i			0.428698i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	227.592			0.885572			0.442786	−	0.896627i	$-0.353990\pi$
$257$	227.592			0.885572			0.442786	+	0.896627i	$0.353990\pi$
$258$		0			0
$259$	−155.824			−0.601638
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−220.422			−0.838105			−0.419053	−	0.907962i	$-0.637638\pi$
$263$	−220.422			−0.838105			−0.419053	+	0.907962i	$0.637638\pi$
$264$		0			0
$265$	−281.265	+	205.860i	−1.06138	+	0.776831i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			252.831i			0.939893i	0.882695	+	0.469946i	$0.155727\pi$
$269$			252.831i			0.939893i	−0.882695	+	0.469946i	$0.844273\pi$
$270$		0			0
$271$	−77.0885			−0.284460			−0.142230	−	0.989834i	$-0.545427\pi$
$271$	−77.0885			−0.284460			−0.142230	+	0.989834i	$0.545427\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	224.787	+	71.3007i	0.817409	+	0.259275i
$276$		0			0
$277$			395.252i			1.42690i	0.700705	+	0.713451i	$0.252869\pi$
$277$			395.252i			1.42690i	−0.700705	+	0.713451i	$0.747131\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			257.575i			0.916635i	0.888788	+	0.458318i	$0.151548\pi$
$281$			257.575i			0.916635i	−0.888788	+	0.458318i	$0.848452\pi$
$282$		0			0
$283$			481.604i			1.70178i	0.525344	+	0.850890i	$0.323937\pi$
$283$			481.604i			1.70178i	−0.525344	+	0.850890i	$0.676063\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−416.189			−1.45013
$288$		0			0
$289$	−252.280			−0.872941
$290$		0			0
$291$		0			0
$292$		0			0
$293$	538.266			1.83709			0.918543	−	0.395321i	$-0.129367\pi$
$293$	538.266			1.83709			0.918543	+	0.395321i	$0.129367\pi$
$294$		0			0
$295$	147.874	+	202.040i	0.501269	+	0.684880i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	78.9395i		−	0.264012i
$300$		0			0
$301$	349.393			1.16077
$302$		0			0
$303$		0			0
$304$		0			0
$305$	374.905	−	274.396i	1.22920	−	0.899658i
$306$		0			0
$307$		−	286.244i		−	0.932389i	−0.884682	−	0.466195i	$-0.845625\pi$
$307$		−	286.244i		−	0.932389i	0.884682	−	0.466195i	$-0.154375\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	426.662i		−	1.37190i	−0.727648	−	0.685951i	$-0.759386\pi$
$311$		−	426.662i		−	1.37190i	0.727648	−	0.685951i	$-0.240614\pi$
$312$		0			0
$313$		−	61.5426i		−	0.196622i	−0.995156	−	0.0983109i	$-0.968656\pi$
$313$		−	61.5426i		−	0.196622i	0.995156	−	0.0983109i	$-0.0313439\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−72.9235			−0.230043			−0.115021	−	0.993363i	$-0.536694\pi$
$317$	−72.9235			−0.230043			−0.115021	+	0.993363i	$0.536694\pi$
$318$		0			0
$319$	200.072			0.627184
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−86.8713			−0.268951
$324$		0			0
$325$	−163.604	−	51.8939i	−0.503397	−	0.159673i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			353.074i			1.07317i
$330$		0			0
$331$	−554.716			−1.67588			−0.837940	−	0.545763i	$-0.816240\pi$
$331$	−554.716			−1.67588			−0.837940	+	0.545763i	$0.816240\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	124.402	+	169.969i	0.371348	+	0.507370i
$336$		0			0
$337$		−	424.454i		−	1.25951i	−0.776794	−	0.629754i	$-0.783156\pi$
$337$		−	424.454i		−	1.25951i	0.776794	−	0.629754i	$-0.216844\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			79.9199i			0.234369i
$342$		0			0
$343$			339.353i			0.989368i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−29.1438			−0.0839880			−0.0419940	−	0.999118i	$-0.513371\pi$
$347$	−29.1438			−0.0839880			−0.0419940	+	0.999118i	$0.513371\pi$
$348$		0			0
$349$	433.957			1.24343			0.621715	−	0.783244i	$-0.286436\pi$
$349$	433.957			1.24343			0.621715	+	0.783244i	$0.286436\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	193.520			0.548215			0.274108	−	0.961699i	$-0.411618\pi$
$353$	193.520			0.548215			0.274108	+	0.961699i	$0.411618\pi$
$354$		0			0
$355$	248.835	+	339.982i	0.700944	+	0.957694i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	72.2543i		−	0.201265i	−0.994924	−	0.100633i	$-0.967913\pi$
$359$		−	72.2543i		−	0.201265i	0.994924	−	0.100633i	$-0.0320867\pi$
$360$		0			0
$361$	−155.483			−0.430700
$362$		0			0
$363$		0			0
$364$		0			0
$365$	321.038	+	438.632i	0.879557	+	1.20173i
$366$		0			0
$367$		−	27.8675i		−	0.0759332i	−0.999279	−	0.0379666i	$-0.987912\pi$
$367$		−	27.8675i		−	0.0759332i	0.999279	−	0.0379666i	$-0.0120880\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	493.005i		−	1.32885i
$372$		0			0
$373$			372.131i			0.997672i	0.866697	+	0.498836i	$0.166239\pi$
$373$			372.131i			0.997672i	−0.866697	+	0.498836i	$0.833761\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−145.616			−0.386249
$378$		0			0
$379$	84.9969			0.224266			0.112133	−	0.993693i	$-0.464232\pi$
$379$	84.9969			0.224266			0.112133	+	0.993693i	$0.464232\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.6137			0.0277119			0.0138560	−	0.999904i	$-0.495589\pi$
$383$	10.6137			0.0277119			0.0138560	+	0.999904i	$0.495589\pi$
$384$		0			0
$385$	−269.166	+	197.005i	−0.699133	+	0.511701i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			30.6939i			0.0789046i	0.999221	+	0.0394523i	$0.0125613\pi$
$389$			30.6939i			0.0789046i	−0.999221	+	0.0394523i	$0.987439\pi$
$390$		0			0
$391$	−69.6748			−0.178196
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−93.8810	+	68.7123i	−0.237673	+	0.173955i
$396$		0			0
$397$		−	322.601i		−	0.812596i	−0.913741	−	0.406298i	$-0.866820\pi$
$397$		−	322.601i		−	0.812596i	0.913741	−	0.406298i	$-0.133180\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			147.729i			0.368400i	0.982889	+	0.184200i	$0.0589695\pi$
$401$			147.729i			0.368400i	−0.982889	+	0.184200i	$0.941031\pi$
$402$		0			0
$403$		−	58.1671i		−	0.144335i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−207.840			−0.510664
$408$		0			0
$409$	−681.238			−1.66562			−0.832810	−	0.553560i	$-0.813269\pi$
$409$	−681.238			−1.66562			−0.832810	+	0.553560i	$0.813269\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−354.137			−0.857476
$414$		0			0
$415$	224.778	−	164.517i	0.541635	−	0.396427i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	428.504i		−	1.02268i	−0.859378	−	0.511341i	$-0.829149\pi$
$419$		−	428.504i		−	1.02268i	0.859378	−	0.511341i	$-0.170851\pi$
$420$		0			0
$421$	659.688			1.56695			0.783477	−	0.621421i	$-0.213444\pi$
$421$	659.688			1.56695			0.783477	+	0.621421i	$0.213444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−45.8033	+	144.403i	−0.107773	+	0.339771i
$426$		0			0
$427$			657.138i			1.53896i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	421.860i		−	0.978794i	−0.872061	−	0.489397i	$-0.837217\pi$
$431$		−	421.860i		−	0.978794i	0.872061	−	0.489397i	$-0.162783\pi$
$432$		0			0
$433$		−	145.766i		−	0.336641i	−0.985732	−	0.168321i	$-0.946166\pi$
$433$		−	145.766i		−	0.336641i	0.985732	−	0.168321i	$-0.0538344\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	164.834			0.377195
$438$		0			0
$439$	193.485			0.440741			0.220371	−	0.975416i	$-0.429273\pi$
$439$	193.485			0.440741			0.220371	+	0.975416i	$0.429273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−548.667			−1.23853			−0.619263	−	0.785184i	$-0.712568\pi$
$443$	−548.667			−1.23853			−0.619263	+	0.785184i	$0.712568\pi$
$444$		0			0
$445$	−216.069	−	295.213i	−0.485548	−	0.663400i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	486.456i		−	1.08342i	−0.840565	−	0.541711i	$-0.817777\pi$
$449$		−	486.456i		−	1.08342i	0.840565	−	0.541711i	$-0.182223\pi$
$450$		0			0
$451$	−555.118			−1.23086
$452$		0			0
$453$		0			0
$454$		0			0
$455$	195.904	−	143.384i	0.430558	−	0.315129i
$456$		0			0
$457$			20.7241i			0.0453482i	0.999743	+	0.0226741i	$0.00721801\pi$
$457$			20.7241i			0.0453482i	−0.999743	+	0.0226741i	$0.992782\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	425.420i		−	0.922819i	−0.887187	−	0.461409i	$-0.847344\pi$
$461$		−	425.420i		−	0.922819i	0.887187	−	0.461409i	$-0.152656\pi$
$462$		0			0
$463$		−	894.704i		−	1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$463$		−	894.704i		−	1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−700.964			−1.50099			−0.750497	−	0.660874i	$-0.770186\pi$
$467$	−700.964			−1.50099			−0.750497	+	0.660874i	$0.770186\pi$
$468$		0			0
$469$	−297.923			−0.635231
$470$		0			0
$471$		0			0
$472$		0			0
$473$	466.025			0.985254
$474$		0			0
$475$	108.360	−	341.623i	0.228126	−	0.719206i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			122.329i			0.255384i	0.991814	+	0.127692i	$0.0407570\pi$
$479$			122.329i			0.255384i	−0.991814	+	0.127692i	$0.959243\pi$
$480$		0			0
$481$	151.270			0.314490
$482$		0			0
$483$		0			0
$484$		0			0
$485$	205.188	+	280.347i	0.423068	+	0.578034i
$486$		0			0
$487$		−	220.179i		−	0.452112i	−0.974114	−	0.226056i	$-0.927417\pi$
$487$		−	220.179i		−	0.452112i	0.974114	−	0.226056i	$-0.0725833\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			670.307i			1.36519i	0.730798	+	0.682594i	$0.239148\pi$
$491$			670.307i			1.36519i	−0.730798	+	0.682594i	$0.760852\pi$
$492$		0			0
$493$			128.525i			0.260701i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−595.924			−1.19904
$498$		0			0
$499$	10.8177			0.0216788			0.0108394	−	0.999941i	$-0.496550\pi$
$499$	10.8177			0.0216788			0.0108394	+	0.999941i	$0.496550\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	299.614			0.595655			0.297828	−	0.954620i	$-0.403738\pi$
$503$	299.614			0.595655			0.297828	+	0.954620i	$0.403738\pi$
$504$		0			0
$505$	−379.079	−	517.932i	−0.750651	−	1.02561i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			467.799i			0.919054i	0.888164	+	0.459527i	$0.151981\pi$
$509$			467.799i			0.919054i	−0.888164	+	0.459527i	$0.848019\pi$
$510$		0			0
$511$	−768.840			−1.50458
$512$		0			0
$513$		0			0
$514$		0			0
$515$	155.631	+	212.638i	0.302197	+	0.412889i
$516$		0			0
$517$			470.935i			0.910899i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	817.672i		−	1.56943i	−0.619858	−	0.784714i	$-0.712810\pi$
$521$		−	817.672i		−	1.56943i	0.619858	−	0.784714i	$-0.287190\pi$
$522$		0			0
$523$		−	128.171i		−	0.245070i	−0.992464	−	0.122535i	$-0.960898\pi$
$523$		−	128.171i		−	0.245070i	0.992464	−	0.122535i	$-0.0391023\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−51.3403			−0.0974199
$528$		0			0
$529$	−396.795			−0.750086
$530$		0			0
$531$		0			0
$532$		0			0
$533$	404.025			0.758020
$534$		0			0
$535$	−25.0238	+	18.3151i	−0.0467734	+	0.0342338i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	9.58220i		−	0.0177777i
$540$		0			0
$541$	988.719			1.82758			0.913788	−	0.406191i	$-0.133143\pi$
$541$	988.719			1.82758			0.913788	+	0.406191i	$0.133143\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	248.001	−	181.514i	0.455048	−	0.333054i
$546$		0			0
$547$		−	219.588i		−	0.401440i	−0.979649	−	0.200720i	$-0.935672\pi$
$547$		−	219.588i		−	0.401440i	0.979649	−	0.200720i	$-0.0643282\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	304.061i		−	0.551835i
$552$		0			0
$553$		−	164.556i		−	0.297569i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−550.582			−0.988478			−0.494239	−	0.869326i	$-0.664553\pi$
$557$	−550.582			−0.988478			−0.494239	+	0.869326i	$0.664553\pi$
$558$		0			0
$559$	−339.181			−0.606764
$560$		0			0
$561$		0			0
$562$		0			0
$563$	499.451			0.887125			0.443562	−	0.896244i	$-0.353714\pi$
$563$	499.451			0.887125			0.443562	+	0.896244i	$0.353714\pi$
$564$		0			0
$565$	−58.2309	+	42.6196i	−0.103063	+	0.0754330i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	570.215i		−	1.00213i	−0.865408	−	0.501067i	$-0.832941\pi$
$569$		−	570.215i		−	1.00213i	0.865408	−	0.501067i	$-0.167059\pi$
$570$		0			0
$571$	−297.528			−0.521065			−0.260533	−	0.965465i	$-0.583898\pi$
$571$	−297.528			−0.521065			−0.260533	+	0.965465i	$0.583898\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	86.9097	−	273.997i	0.151147	−	0.476517i
$576$		0			0
$577$		−	856.924i		−	1.48514i	−0.669770	−	0.742569i	$-0.733607\pi$
$577$		−	856.924i		−	1.48514i	0.669770	−	0.742569i	$-0.266393\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			393.994i			0.678131i
$582$		0			0
$583$		−	657.577i		−	1.12792i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−285.152			−0.485779			−0.242890	−	0.970054i	$-0.578095\pi$
$587$	−285.152			−0.485779			−0.242890	+	0.970054i	$0.578095\pi$
$588$		0			0
$589$	121.459			0.206212
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1110.27			−1.87230			−0.936148	−	0.351607i	$-0.885635\pi$
$593$	−1110.27			−1.87230			−0.936148	+	0.351607i	$0.885635\pi$
$594$		0			0
$595$	−126.555	−	172.912i	−0.212698	−	0.290608i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			290.549i			0.485056i	0.970144	+	0.242528i	$0.0779767\pi$
$599$			290.549i			0.485056i	−0.970144	+	0.242528i	$0.922023\pi$
$600$		0			0
$601$	−840.436			−1.39840			−0.699198	−	0.714928i	$-0.746460\pi$
$601$	−840.436			−1.39840			−0.699198	+	0.714928i	$0.746460\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	129.189	−	94.5544i	0.213535	−	0.156288i
$606$		0			0
$607$		−	225.751i		−	0.371912i	−0.982558	−	0.185956i	$-0.940462\pi$
$607$		−	225.751i		−	0.371912i	0.982558	−	0.185956i	$-0.0595382\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	342.755i		−	0.560973i
$612$		0			0
$613$			395.174i			0.644656i	0.946628	+	0.322328i	$0.104465\pi$
$613$			395.174i			0.644656i	−0.946628	+	0.322328i	$0.895535\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−559.277			−0.906445			−0.453223	−	0.891397i	$-0.649726\pi$
$617$	−559.277			−0.906445			−0.453223	+	0.891397i	$0.649726\pi$
$618$		0			0
$619$	571.556			0.923354			0.461677	−	0.887048i	$-0.347248\pi$
$619$	571.556			0.923354			0.461677	+	0.887048i	$0.347248\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	517.453			0.830583
$624$		0			0
$625$	−510.733	−	360.245i	−0.817173	−	0.576392i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	133.516i		−	0.212267i
$630$		0			0
$631$	−154.051			−0.244138			−0.122069	−	0.992522i	$-0.538953\pi$
$631$	−154.051			−0.244138			−0.122069	+	0.992522i	$0.538953\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	641.516	+	876.498i	1.01026	+	1.38031i
$636$		0			0
$637$			6.97409i			0.0109483i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	799.864i		−	1.24784i	−0.781489	−	0.623919i	$-0.785540\pi$
$641$		−	799.864i		−	1.24784i	0.781489	−	0.623919i	$-0.214460\pi$
$642$		0			0
$643$		−	562.528i		−	0.874849i	−0.899255	−	0.437425i	$-0.855891\pi$
$643$		−	562.528i		−	0.874849i	0.899255	−	0.437425i	$-0.144109\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1160.79			−1.79412			−0.897059	−	0.441910i	$-0.854301\pi$
$647$	−1160.79			−1.79412			−0.897059	+	0.441910i	$0.854301\pi$
$648$		0			0
$649$	−472.353			−0.727817
$650$		0			0
$651$		0			0
$652$		0			0
$653$	812.354			1.24403			0.622017	−	0.783004i	$-0.286313\pi$
$653$	812.354			1.24403			0.622017	+	0.783004i	$0.286313\pi$
$654$		0			0
$655$	343.268	+	469.004i	0.524073	+	0.716037i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	552.026i		−	0.837673i	−0.908062	−	0.418836i	$-0.862438\pi$
$659$		−	552.026i		−	0.837673i	0.908062	−	0.418836i	$-0.137562\pi$
$660$		0			0
$661$	480.917			0.727559			0.363780	−	0.931485i	$-0.381486\pi$
$661$	480.917			0.727559			0.363780	+	0.931485i	$0.381486\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	299.401	+	409.068i	0.450226	+	0.615141i
$666$		0			0
$667$		−	243.871i		−	0.365624i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			876.499i			1.30626i
$672$		0			0
$673$		−	462.464i		−	0.687167i	−0.939122	−	0.343584i	$-0.888359\pi$
$673$		−	462.464i		−	0.687167i	0.939122	−	0.343584i	$-0.111641\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1136.52			−1.67876			−0.839381	−	0.543544i	$-0.817082\pi$
$677$	−1136.52			−1.67876			−0.839381	+	0.543544i	$0.817082\pi$
$678$		0			0
$679$	−491.395			−0.723704
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1329.13			1.94601			0.973006	−	0.230779i	$-0.0741274\pi$
$683$	1329.13			1.94601			0.973006	+	0.230779i	$0.0741274\pi$
$684$		0			0
$685$	−530.799	+	388.496i	−0.774888	+	0.567147i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			478.596i			0.694624i
$690$		0			0
$691$	−1131.72			−1.63781			−0.818903	−	0.573932i	$-0.805417\pi$
$691$	−1131.72			−1.63781			−0.818903	+	0.573932i	$0.805417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	995.552	−	728.653i	1.43245	−	1.04842i
$696$		0			0
$697$		−	356.606i		−	0.511630i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	279.897i		−	0.399282i	−0.979869	−	0.199641i	$-0.936022\pi$
$701$		−	279.897i		−	0.399282i	0.979869	−	0.199641i	$-0.0639776\pi$
$702$		0			0
$703$			315.868i			0.449314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	907.838			1.28407
$708$		0			0
$709$	−39.9326			−0.0563225			−0.0281612	−	0.999603i	$-0.508965\pi$
$709$	−39.9326			−0.0563225			−0.0281612	+	0.999603i	$0.508965\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	97.4158			0.136628
$714$		0			0
$715$	261.299	−	191.247i	0.365454	−	0.267478i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			507.415i			0.705724i	0.935675	+	0.352862i	$0.114791\pi$
$719$			507.415i			0.705724i	−0.935675	+	0.352862i	$0.885209\pi$
$720$		0			0
$721$	−372.714			−0.516941
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−505.429	−	160.318i	−0.697144	−	0.221128i
$726$		0			0
$727$			771.057i			1.06060i	0.847810	+	0.530300i	$0.177921\pi$
$727$			771.057i			1.06060i	−0.847810	+	0.530300i	$0.822079\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			299.373i			0.409539i
$732$		0			0
$733$		−	1002.00i		−	1.36698i	−0.729960	−	0.683490i	$-0.760461\pi$
$733$		−	1002.00i		−	1.36698i	0.729960	−	0.683490i	$-0.239539\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−397.374			−0.539178
$738$		0			0
$739$	−277.168			−0.375058			−0.187529	−	0.982259i	$-0.560048\pi$
$739$	−277.168			−0.375058			−0.187529	+	0.982259i	$0.560048\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	550.164			0.740463			0.370231	−	0.928940i	$-0.379278\pi$
$743$	550.164			0.740463			0.370231	+	0.928940i	$0.379278\pi$
$744$		0			0
$745$	−524.537	−	716.671i	−0.704077	−	0.961974i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	43.8620i		−	0.0585607i
$750$		0			0
$751$	−990.738			−1.31923			−0.659613	−	0.751606i	$-0.729280\pi$
$751$	−990.738			−1.31923			−0.659613	+	0.751606i	$0.729280\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−688.345	+	503.805i	−0.911715	+	0.667292i
$756$		0			0
$757$			970.548i			1.28210i	0.767500	+	0.641049i	$0.221500\pi$
$757$			970.548i			1.28210i	−0.767500	+	0.641049i	$0.778500\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	324.468i		−	0.426371i	−0.977012	−	0.213186i	$-0.931616\pi$
$761$		−	324.468i		−	0.426371i	0.977012	−	0.213186i	$-0.0683839\pi$
$762$		0			0
$763$			434.700i			0.569725i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	343.787			0.448223
$768$		0			0
$769$	−927.886			−1.20661			−0.603307	−	0.797509i	$-0.706151\pi$
$769$	−927.886			−1.20661			−0.603307	+	0.797509i	$0.706151\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−214.716			−0.277769			−0.138885	−	0.990309i	$-0.544352\pi$
$773$	−214.716			−0.277769			−0.138885	+	0.990309i	$0.544352\pi$
$774$		0			0
$775$	64.0400	−	201.897i	0.0826322	−	0.260512i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			843.647i			1.08299i
$780$		0			0
$781$	−794.851			−1.01774
$782$		0			0
$783$		0			0
$784$		0			0
$785$	58.2366	+	79.5681i	0.0741867	+	0.101361i
$786$		0			0
$787$		−	607.914i		−	0.772444i	−0.922406	−	0.386222i	$-0.873780\pi$
$787$		−	607.914i		−	0.772444i	0.922406	−	0.386222i	$-0.126220\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	102.068i		−	0.129036i
$792$		0			0
$793$		−	637.931i		−	0.804453i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−683.251			−0.857278			−0.428639	−	0.903476i	$-0.641007\pi$
$797$	−683.251			−0.857278			−0.428639	+	0.903476i	$0.641007\pi$
$798$		0			0
$799$	−302.527			−0.378632
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1025.49			−1.27707
$804$		0			0
$805$	240.133	+	328.092i	0.298302	+	0.407567i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	593.776i		−	0.733962i	−0.930228	−	0.366981i	$-0.880391\pi$
$809$		−	593.776i		−	0.733962i	0.930228	−	0.366981i	$-0.119609\pi$
$810$		0			0
$811$	−176.686			−0.217862			−0.108931	−	0.994049i	$-0.534743\pi$
$811$	−176.686			−0.217862			−0.108931	+	0.994049i	$0.534743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	606.255	+	828.322i	0.743872	+	1.01635i
$816$		0			0
$817$		−	708.247i		−	0.866887i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1098.29i			1.33775i	0.743376	+	0.668874i	$0.233224\pi$
$821$			1098.29i			1.33775i	−0.743376	+	0.668874i	$0.766776\pi$
$822$		0			0
$823$			671.186i			0.815536i	0.913086	+	0.407768i	$0.133693\pi$
$823$			671.186i			0.815536i	−0.913086	+	0.407768i	$0.866307\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−122.072			−0.147608			−0.0738042	−	0.997273i	$-0.523514\pi$
$827$	−122.072			−0.147608			−0.0738042	+	0.997273i	$0.523514\pi$
$828$		0			0
$829$	409.266			0.493687			0.246843	−	0.969055i	$-0.420607\pi$
$829$	409.266			0.493687			0.246843	+	0.969055i	$0.420607\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.15558			0.00738965
$834$		0			0
$835$	313.413	−	229.389i	0.375345	−	0.274718i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1404.60i			1.67414i	0.547094	+	0.837071i	$0.315734\pi$
$839$			1404.60i			1.67414i	−0.547094	+	0.837071i	$0.684266\pi$
$840$		0			0
$841$	391.143			0.465093
$842$		0			0
$843$		0			0
$844$		0			0
$845$	491.697	−	359.877i	0.581890	−	0.425890i
$846$		0			0
$847$			226.444i			0.267348i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			253.340i			0.297697i
$852$		0			0
$853$		−	650.887i		−	0.763056i	−0.924357	−	0.381528i	$-0.875398\pi$
$853$		−	650.887i		−	0.763056i	0.924357	−	0.381528i	$-0.124602\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−609.229			−0.710885			−0.355443	−	0.934698i	$-0.615670\pi$
$857$	−609.229			−0.710885			−0.355443	+	0.934698i	$0.615670\pi$
$858$		0			0
$859$	−643.278			−0.748868			−0.374434	−	0.927254i	$-0.622163\pi$
$859$	−643.278			−0.748868			−0.374434	+	0.927254i	$0.622163\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	515.366			0.597180			0.298590	−	0.954382i	$-0.403484\pi$
$863$	515.366			0.597180			0.298590	+	0.954382i	$0.403484\pi$
$864$		0			0
$865$	144.024	−	105.413i	0.166502	−	0.121864i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	219.487i		−	0.252574i
$870$		0			0
$871$	289.216			0.332050
$872$		0			0
$873$		0			0
$874$		0			0
$875$	837.839	−	281.998i	0.957530	−	0.322284i
$876$		0			0
$877$			1066.77i			1.21639i	0.793789	+	0.608193i	$0.208105\pi$
$877$			1066.77i			1.21639i	−0.793789	+	0.608193i	$0.791895\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1275.90i			1.44824i	0.689673	+	0.724121i	$0.257754\pi$
$881$			1275.90i			1.44824i	−0.689673	+	0.724121i	$0.742246\pi$
$882$		0			0
$883$		−	770.113i		−	0.872156i	−0.899909	−	0.436078i	$-0.856367\pi$
$883$		−	770.113i		−	0.872156i	0.899909	−	0.436078i	$-0.143633\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−867.870			−0.978433			−0.489216	−	0.872162i	$-0.662717\pi$
$887$	−867.870			−0.978433			−0.489216	+	0.872162i	$0.662717\pi$
$888$		0			0
$889$	−1536.34			−1.72816
$890$		0			0
$891$		0			0
$892$		0			0
$893$	715.708			0.801465
$894$		0			0
$895$	−815.830	−	1114.66i	−0.911542	−	1.24543i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	179.698i		−	0.199886i
$900$		0			0
$901$	422.425			0.468840
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1160.23	−	849.179i	1.28202	−	0.938319i
$906$		0			0
$907$		−	661.261i		−	0.729063i	−0.931191	−	0.364532i	$-0.881229\pi$
$907$		−	661.261i		−	0.729063i	0.931191	−	0.364532i	$-0.118771\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			912.077i			1.00118i	0.865684	+	0.500591i	$0.166884\pi$
$911$			912.077i			1.00118i	−0.865684	+	0.500591i	$0.833116\pi$
$912$		0			0
$913$			525.515i			0.575591i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−822.076			−0.896484
$918$		0			0
$919$	27.1804			0.0295760			0.0147880	−	0.999891i	$-0.495293\pi$
$919$	27.1804			0.0295760			0.0147880	+	0.999891i	$0.495293\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	578.506			0.626768
$924$		0			0
$925$	525.054	+	166.543i	0.567626	+	0.180046i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	417.565i		−	0.449478i	−0.974419	−	0.224739i	$-0.927847\pi$
$929$		−	417.565i		−	0.449478i	0.974419	−	0.224739i	$-0.0721530\pi$
$930$		0			0
$931$	−14.5627			−0.0156419
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−168.801	−	230.632i	−0.180536	−	0.246665i
$936$		0			0
$937$			1120.89i			1.19625i	0.801403	+	0.598124i	$0.204087\pi$
$937$			1120.89i			1.19625i	−0.801403	+	0.598124i	$0.795913\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	119.014i		−	0.126477i	−0.997998	−	0.0632383i	$-0.979857\pi$
$941$		−	119.014i		−	0.126477i	0.997998	−	0.0632383i	$-0.0201428\pi$
$942$		0			0
$943$			676.643i			0.717543i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1133.60			−1.19705			−0.598523	−	0.801106i	$-0.704245\pi$
$947$	−1133.60			−1.19705			−0.598523	+	0.801106i	$0.704245\pi$
$948$		0			0
$949$	746.369			0.786479
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1505.57			−1.57982			−0.789909	−	0.613224i	$-0.789872\pi$
$953$	−1505.57			−1.57982			−0.789909	+	0.613224i	$0.789872\pi$
$954$		0			0
$955$	187.423	+	256.074i	0.196254	+	0.268141i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	930.390i		−	0.970167i
$960$		0			0
$961$	−889.219			−0.925305
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−703.316	−	960.934i	−0.728825	−	0.995787i
$966$		0			0
$967$		−	458.952i		−	0.474615i	−0.971435	−	0.237307i	$-0.923735\pi$
$967$		−	458.952i		−	0.474615i	0.971435	−	0.237307i	$-0.0762649\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	189.181i		−	0.194831i	−0.995244	−	0.0974155i	$-0.968942\pi$
$971$		−	189.181i		−	0.194831i	0.995244	−	0.0974155i	$-0.0310576\pi$
$972$		0			0
$973$			1745.02i			1.79344i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−86.2776			−0.0883087			−0.0441543	−	0.999025i	$-0.514059\pi$
$977$	−86.2776			−0.0883087			−0.0441543	+	0.999025i	$0.514059\pi$
$978$		0			0
$979$	690.186			0.704991
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−278.686			−0.283505			−0.141753	−	0.989902i	$-0.545274\pi$
$983$	−278.686			−0.283505			−0.141753	+	0.989902i	$0.545274\pi$
$984$		0			0
$985$	−531.689	+	389.147i	−0.539786	+	0.395074i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	568.047i		−	0.574365i
$990$		0			0
$991$	−666.594			−0.672648			−0.336324	−	0.941746i	$-0.609184\pi$
$991$	−666.594			−0.672648			−0.336324	+	0.941746i	$0.609184\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1252.62	−	916.801i	1.25891	−	0.921408i
$996$		0			0
$997$		−	1565.47i		−	1.57018i	−0.619383	−	0.785089i	$-0.712617\pi$
$997$		−	1565.47i		−	1.57018i	0.619383	−	0.785089i	$-0.287383\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.o.1889.9		12
3.2	odd	2		2160.3.c.p.1889.4		12
4.3	odd	2		1080.3.c.a.809.9	✓	12
5.4	even	2		2160.3.c.p.1889.3		12
12.11	even	2		1080.3.c.b.809.4	yes	12
15.14	odd	2	inner	2160.3.c.o.1889.10		12
20.19	odd	2		1080.3.c.b.809.3	yes	12
60.59	even	2		1080.3.c.a.809.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.a.809.9	✓	12	4.3	odd	2
1080.3.c.a.809.10	yes	12	60.59	even	2
1080.3.c.b.809.3	yes	12	20.19	odd	2
1080.3.c.b.809.4	yes	12	12.11	even	2
2160.3.c.o.1889.9		12	1.1	even	1	trivial
2160.3.c.o.1889.10		12	15.14	odd	2	inner
2160.3.c.p.1889.3		12	5.4	even	2
2160.3.c.p.1889.4		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.03477 - 2.95308i) q^{5} +7.07219i q^{7} +9.43298i q^{11} -6.86548i q^{13} -6.05971 q^{17} +14.3359 q^{19} +11.4980 q^{23} +(7.55866 - 23.8300i) q^{25} -21.2098i q^{29} +8.47239 q^{31} +(20.8847 + 28.5346i) q^{35} +22.0334i q^{37} +58.8487i q^{41} -49.4038i q^{43} +49.9243 q^{47} -1.01582 q^{49} -69.7104 q^{53} +(27.8563 + 38.0599i) q^{55} +50.0747i q^{59} +92.9186 q^{61} +(-20.2743 - 27.7006i) q^{65} +42.1261i q^{67} +84.2630i q^{71} +108.713i q^{73} -66.7118 q^{77} -23.2680 q^{79} +55.7104 q^{83} +(-24.4495 + 17.8948i) q^{85} -73.1674i q^{89} +48.5540 q^{91} +(57.8419 - 42.3349i) q^{95} +69.4827i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{5} + 6 q^{25} - 12 q^{31} + 30 q^{35} - 48 q^{47} - 24 q^{49} + 60 q^{53} - 30 q^{55} - 120 q^{61} + 108 q^{65} - 204 q^{77} - 72 q^{79} + 84 q^{83} - 84 q^{85} - 48 q^{91} + 96 q^{95}+O(q^{100}) \) 12 * q - 6 * q^5 + 6 * q^25 - 12 * q^31 + 30 * q^35 - 48 * q^47 - 24 * q^49 + 60 * q^53 - 30 * q^55 - 120 * q^61 + 108 * q^65 - 204 * q^77 - 72 * q^79 + 84 * q^83 - 84 * q^85 - 48 * q^91 + 96 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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